In a small bowl, combine the bread crumbs and butter and set aside. Lightly grease an 8X8-inch baking dish and set aside.

Preheat the oven to 400 degrees and make sure the oven rack is placed in the middle of the oven. In a large pot, bring the water for the pasta to a boil. Stir in some salt and the broccoli. Cook the broccoli for 1 minute, until it is bright green. Quickly remove the broccoli with a slotted spoon to a plate. Return the water to a boil and add the pasta and cook until al dente, about 1-2 minutes less than how you would eat it normally (it will cook a bit more in the oven while baking). Drain the pasta in a colander and toss with 1/2 tablespoon olive oil. Leave it in the colander and set it aside.

Wipe the pot dry. Add the remaining 1/2 tablespoon oil and return to medium heat until shimmering. Add the onion and cook until softened and beginning to brown, about 5 minutes. Stir in the garlic and thyme; cook until fragrant, about 30 seconds, stirring constantly. Add the flour and cook, stirring constantly, until golden, about 1 minute. Slowly whisk in the broth and cream; bring to a simmer, whisking often. Add the chicken and cook, stirring occasionally, until cooked through, about 6-7 minutes. Stir in the sun-dried tomatoes.

Add the cooked pasta and broccoli to the sauce; stir to combine. Transfer the pasta mixture to the prepared baking dish and sprinkle with the cheese and the bread crumb topping. Bake until the casserole is bubbling and the crumbs are lightly browned, about 15 minutes. Serve immediately.

*The casserole can be prepared in advance with the following steps: after the broccoli is removed from the boiling water, plunge it into a bowl of ice water and cool completely. Assemble the casserole following the directions in the recipe. After the casserole is spread in the prepared baking pan, let it cool for 5-10 minutes before covering with plastic wrap and refrigerating for up to 8 hours. To bake, preheat the oven to 375 and bake uncovered for 30 minutes.